musical, physical, and mathematical intervals the 2010 ...intervals the 2010 leonard sulski lecture...
TRANSCRIPT
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Musical, Physical,and Mathematical Intervals
The 2010 Leonard Sulski LectureCollege of the Holy Cross
Rick Miranda, Colorado State University
April 12, 2010
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Outline
The Physics of Sound
Length (or Frequency) Ratios Between Notes
Fretting A Guitar
Geometrical Approximations
Arithmetic Approximations
Vincenzo Galilei
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Vibrating Strings
When a string vibrates, its basic pitch(the frequency of the sound wave generated)is determined by
I the composition of the string (thickness, material, etc.)
I the tension of the string
I the length of the string.
Ancient Scientists knew that Frequency and Length areinversely proportional:
Frequency =(constant)
Length
(Although they didn’t really know much about Frequency...)
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Vibrating Strings
When a string vibrates, its basic pitch(the frequency of the sound wave generated)is determined by
I the composition of the string (thickness, material, etc.)
I the tension of the string
I the length of the string.
Ancient Scientists knew that Frequency and Length areinversely proportional:
Frequency =(constant)
Length
(Although they didn’t really know much about Frequency...)
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Pythagorean Intervals
The Pythagorean School refined this one step further.They considered two notes together: Harmony
They noticed that the most pleasing harmonieswere produced by Frequencies(actually, they used Lengths as the measure)which were in ratios of small integers:
I Octave: (e.g. middle C to high C): 1 - to - 2
I Fifth: (e.g. C to G): 2 - to - 3
I Fourth: (e.g. C to F): 3 - to - 4
I Etc.
There is a “Resonance” reason for this:the wave form produced by adding waves with these ratiosare simpler, less discordant (even visually)
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Pythagorean Intervals
The Pythagorean School refined this one step further.They considered two notes together: HarmonyThey noticed that the most pleasing harmonieswere produced by Frequencies(actually, they used Lengths as the measure)which were in ratios of small integers:
I Octave: (e.g. middle C to high C): 1 - to - 2
I Fifth: (e.g. C to G): 2 - to - 3
I Fourth: (e.g. C to F): 3 - to - 4
I Etc.
There is a “Resonance” reason for this:the wave form produced by adding waves with these ratiosare simpler, less discordant (even visually)
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Pythagorean Intervals
The Pythagorean School refined this one step further.They considered two notes together: HarmonyThey noticed that the most pleasing harmonieswere produced by Frequencies(actually, they used Lengths as the measure)which were in ratios of small integers:
I Octave: (e.g. middle C to high C): 1 - to - 2
I Fifth: (e.g. C to G): 2 - to - 3
I Fourth: (e.g. C to F): 3 - to - 4
I Etc.
There is a “Resonance” reason for this:the wave form produced by adding waves with these ratiosare simpler, less discordant (even visually)
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Pythagorean Intervals
The Pythagorean School refined this one step further.They considered two notes together: HarmonyThey noticed that the most pleasing harmonieswere produced by Frequencies(actually, they used Lengths as the measure)which were in ratios of small integers:
I Octave: (e.g. middle C to high C): 1 - to - 2
I Fifth: (e.g. C to G): 2 - to - 3
I Fourth: (e.g. C to F): 3 - to - 4
I Etc.
There is a “Resonance” reason for this:the wave form produced by adding waves with these ratiosare simpler, less discordant (even visually)
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Octaves: Ratio = 2
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Dissonance: Ratio = 1.8
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
More Dissonance: Ratio = 1.6
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Fifths: Ratio = 1.5
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Fifths: Ratio =√
2
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Fifths and Fourths seem consistent, at least for a while:
F G C F G C32
43 1
34
23
12
I Octaves: F - to - F ratio = 34/32 =
12 .
I Also G - to - G ratio = 23/43 =
12 .
I Fifths: F - to - C ratio = 1/32 =23 ;
I Also upper F - to - C ratio = 12/34 =
23 .
I Fourths: G - to - C ratio = 1/43 =12/
23 =
34 .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Fifths and Fourths seem consistent, at least for a while:
F G C F G C32
43 1
34
23
12
I Octaves: F - to - F ratio = 34/32 =
12 .
I Also G - to - G ratio = 23/43 =
12 .
I Fifths: F - to - C ratio = 1/32 =23 ;
I Also upper F - to - C ratio = 12/34 =
23 .
I Fourths: G - to - C ratio = 1/43 =12/
23 =
34 .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
More Notes To The Scale
This Pythagorean model works well for scales that onlyinvolve C’s, F’s, and G’s.Let’s try to add a few more notes to the scale.
A fifth above lower G is a D, and the Length should be
23 ∗
43 =
89 .
A fifth above that D is an A, and the Length should be
23 ∗
89 =
1627 .
A fourth below that A is an E, and the Length should be
43 ∗
1627 =
6481 .
A fifth above that E is an B, and the Length should be
23 ∗
6481 =
128243 .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
More Notes To The Scale
This Pythagorean model works well for scales that onlyinvolve C’s, F’s, and G’s.Let’s try to add a few more notes to the scale.A fifth above lower G is a D, and the Length should be
23 ∗
43 =
89 .
A fifth above that D is an A, and the Length should be
23 ∗
89 =
1627 .
A fourth below that A is an E, and the Length should be
43 ∗
1627 =
6481 .
A fifth above that E is an B, and the Length should be
23 ∗
6481 =
128243 .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
More Notes To The Scale
This Pythagorean model works well for scales that onlyinvolve C’s, F’s, and G’s.Let’s try to add a few more notes to the scale.A fifth above lower G is a D, and the Length should be
23 ∗
43 =
89 .
A fifth above that D is an A, and the Length should be
23 ∗
89 =
1627 .
A fourth below that A is an E, and the Length should be
43 ∗
1627 =
6481 .
A fifth above that E is an B, and the Length should be
23 ∗
6481 =
128243 .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
More Notes To The Scale
This Pythagorean model works well for scales that onlyinvolve C’s, F’s, and G’s.Let’s try to add a few more notes to the scale.A fifth above lower G is a D, and the Length should be
23 ∗
43 =
89 .
A fifth above that D is an A, and the Length should be
23 ∗
89 =
1627 .
A fourth below that A is an E, and the Length should be
43 ∗
1627 =
6481 .
A fifth above that E is an B, and the Length should be
23 ∗
6481 =
128243 .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
More Notes To The Scale
This Pythagorean model works well for scales that onlyinvolve C’s, F’s, and G’s.Let’s try to add a few more notes to the scale.A fifth above lower G is a D, and the Length should be
23 ∗
43 =
89 .
A fifth above that D is an A, and the Length should be
23 ∗
89 =
1627 .
A fourth below that A is an E, and the Length should be
43 ∗
1627 =
6481 .
A fifth above that E is an B, and the Length should be
23 ∗
6481 =
128243 .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Il Diavolo In Musica
This gives the ”white notes on the piano” scale:
C D E F G A B C1 89
6481
34
23
1627
128243
12
89
89
243256
89
89
89
243256
This leads to the scheme of:
I Whole Note = ratio of 8/9
I Half Note = ratio of 243/256
And the Problem (”Il Diavolo”) is thatTwo Half Notes should equal a Whole note; but (243256)
2 6= 89 !Close though:
(243
256)2 = .901016235 while
8
9= .888888888
About One Point Three Percent Off. ”Pythagorean Comma”
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Il Diavolo In Musica
This gives the ”white notes on the piano” scale:
C D E F G A B C1 89
6481
34
23
1627
128243
12
89
89
243256
89
89
89
243256
This leads to the scheme of:
I Whole Note = ratio of 8/9
I Half Note = ratio of 243/256
And the Problem (”Il Diavolo”) is thatTwo Half Notes should equal a Whole note; but (243256)
2 6= 89 !Close though:
(243
256)2 = .901016235 while
8
9= .888888888
About One Point Three Percent Off. ”Pythagorean Comma”
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Il Diavolo In Musica
This gives the ”white notes on the piano” scale:
C D E F G A B C1 89
6481
34
23
1627
128243
12
89
89
243256
89
89
89
243256
This leads to the scheme of:
I Whole Note = ratio of 8/9
I Half Note = ratio of 243/256
And the Problem (”Il Diavolo”) is thatTwo Half Notes should equal a Whole note; but (243256)
2 6= 89 !
Close though:
(243
256)2 = .901016235 while
8
9= .888888888
About One Point Three Percent Off. ”Pythagorean Comma”
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Il Diavolo In Musica
This gives the ”white notes on the piano” scale:
C D E F G A B C1 89
6481
34
23
1627
128243
12
89
89
243256
89
89
89
243256
This leads to the scheme of:
I Whole Note = ratio of 8/9
I Half Note = ratio of 243/256
And the Problem (”Il Diavolo”) is thatTwo Half Notes should equal a Whole note; but (243256)
2 6= 89 !Close though:
(243
256)2 = .901016235 while
8
9= .888888888
About One Point Three Percent Off. ”Pythagorean Comma”
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
12 Note Scales?
It gets worse if you try to make a full 12-note scale(including the ’black notes on the piano’).
Twelve Fifths (C - to - G)should be the same as Seven Octaves.But (2/3)12 6= (1/2)7: This is equivalent to524288 = 219 6= 312 = 531441. (1.3% off...)Is there any way to recover from this?Not really.No system of ratios enjoys the following properties:
I The ratio of all half notes (or all whole notes, or...) arethe same
I The ratio of octaves is 1 - to - 2
I The ratio of fifths is 2 - to - 3.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
12 Note Scales?
It gets worse if you try to make a full 12-note scale(including the ’black notes on the piano’).Twelve Fifths (C - to - G)should be the same as Seven Octaves.
But (2/3)12 6= (1/2)7: This is equivalent to524288 = 219 6= 312 = 531441. (1.3% off...)Is there any way to recover from this?Not really.No system of ratios enjoys the following properties:
I The ratio of all half notes (or all whole notes, or...) arethe same
I The ratio of octaves is 1 - to - 2
I The ratio of fifths is 2 - to - 3.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
12 Note Scales?
It gets worse if you try to make a full 12-note scale(including the ’black notes on the piano’).Twelve Fifths (C - to - G)should be the same as Seven Octaves.But (2/3)12 6= (1/2)7: This is equivalent to524288 = 219 6= 312 = 531441. (1.3% off...)
Is there any way to recover from this?Not really.No system of ratios enjoys the following properties:
I The ratio of all half notes (or all whole notes, or...) arethe same
I The ratio of octaves is 1 - to - 2
I The ratio of fifths is 2 - to - 3.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
12 Note Scales?
It gets worse if you try to make a full 12-note scale(including the ’black notes on the piano’).Twelve Fifths (C - to - G)should be the same as Seven Octaves.But (2/3)12 6= (1/2)7: This is equivalent to524288 = 219 6= 312 = 531441. (1.3% off...)Is there any way to recover from this?
Not really.No system of ratios enjoys the following properties:
I The ratio of all half notes (or all whole notes, or...) arethe same
I The ratio of octaves is 1 - to - 2
I The ratio of fifths is 2 - to - 3.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
12 Note Scales?
It gets worse if you try to make a full 12-note scale(including the ’black notes on the piano’).Twelve Fifths (C - to - G)should be the same as Seven Octaves.But (2/3)12 6= (1/2)7: This is equivalent to524288 = 219 6= 312 = 531441. (1.3% off...)Is there any way to recover from this?Not really.No system of ratios enjoys the following properties:
I The ratio of all half notes (or all whole notes, or...) arethe same
I The ratio of octaves is 1 - to - 2
I The ratio of fifths is 2 - to - 3.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Suppose you set the ratio of half note Frequencies to be afixed number HF .
There are 12 half-notes in an octave.So you then need (HF )
12 = 2.This is a number: HF =
12√
2 = 1.059463094 · · ·The corresponding ratio of Lengths would then be
HL = 1/HF = 1/12√
2 = 1/1.059463094 = .943874313 · · ·
Compare this with the Pythagorean half-note length ratio of
243
256= .94921875 · · ·
(These differ by about a half of one percent.)Whole note ratios are then
L2F = .890898 · · · (8
9= .888888 · · · )
These differ by about a fifth of one percent.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Suppose you set the ratio of half note Frequencies to be afixed number HF .There are 12 half-notes in an octave.So you then need (HF )
12 = 2.
This is a number: HF =12√
2 = 1.059463094 · · ·The corresponding ratio of Lengths would then be
HL = 1/HF = 1/12√
2 = 1/1.059463094 = .943874313 · · ·
Compare this with the Pythagorean half-note length ratio of
243
256= .94921875 · · ·
(These differ by about a half of one percent.)Whole note ratios are then
L2F = .890898 · · · (8
9= .888888 · · · )
These differ by about a fifth of one percent.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Suppose you set the ratio of half note Frequencies to be afixed number HF .There are 12 half-notes in an octave.So you then need (HF )
12 = 2.This is a number: HF =
12√
2 = 1.059463094 · · ·The corresponding ratio of Lengths would then be
HL = 1/HF = 1/12√
2 = 1/1.059463094 = .943874313 · · ·
Compare this with the Pythagorean half-note length ratio of
243
256= .94921875 · · ·
(These differ by about a half of one percent.)Whole note ratios are then
L2F = .890898 · · · (8
9= .888888 · · · )
These differ by about a fifth of one percent.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Suppose you set the ratio of half note Frequencies to be afixed number HF .There are 12 half-notes in an octave.So you then need (HF )
12 = 2.This is a number: HF =
12√
2 = 1.059463094 · · ·The corresponding ratio of Lengths would then be
HL = 1/HF = 1/12√
2 = 1/1.059463094 = .943874313 · · ·
Compare this with the Pythagorean half-note length ratio of
243
256= .94921875 · · ·
(These differ by about a half of one percent.)
Whole note ratios are then
L2F = .890898 · · · (8
9= .888888 · · · )
These differ by about a fifth of one percent.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Suppose you set the ratio of half note Frequencies to be afixed number HF .There are 12 half-notes in an octave.So you then need (HF )
12 = 2.This is a number: HF =
12√
2 = 1.059463094 · · ·The corresponding ratio of Lengths would then be
HL = 1/HF = 1/12√
2 = 1/1.059463094 = .943874313 · · ·
Compare this with the Pythagorean half-note length ratio of
243
256= .94921875 · · ·
(These differ by about a half of one percent.)Whole note ratios are then
L2F = .890898 · · · (8
9= .888888 · · · )
These differ by about a fifth of one percent.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Compare the scales:Note C D E F G A B CPyth. 1 .8889 .7901 .75 .6667 .5926 .5267 .5Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5
Percent 0 −0.226 −0.451 −0.113 −0.113 −0.338 −0.563 0
Minor Second and Major Third are the worst.
It is said that a musician’s ear can tolerate about 14 = 0.25percent before running screaming from the room.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Compare the scales:Note C D E F G A B CPyth. 1 .8889 .7901 .75 .6667 .5926 .5267 .5Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5
Percent 0 −0.226 −0.451 −0.113 −0.113 −0.338 −0.563 0
Minor Second and Major Third are the worst.It is said that a musician’s ear can tolerate about 14 = 0.25percent
before running screaming from the room.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Compare the scales:Note C D E F G A B CPyth. 1 .8889 .7901 .75 .6667 .5926 .5267 .5Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5
Percent 0 −0.226 −0.451 −0.113 −0.113 −0.338 −0.563 0
Minor Second and Major Third are the worst.It is said that a musician’s ear can tolerate about 14 = 0.25percent before running screaming from the room.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Fretting A Guitar
Neck Octave Bridge
↑1
↑12√.5
↑0
↑.5
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.Possible Solutions?
I Play the Violin instead
I Use a Computer (not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.
It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.Possible Solutions?
I Play the Violin instead
I Use a Computer (not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.
Possible Solutions?
I Play the Violin instead
I Use a Computer (not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.Possible Solutions?
I Play the Violin instead
I Use a Computer (not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.Possible Solutions?
I Play the Violin instead
I Use a Computer (not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.Possible Solutions?
I Play the Violin instead
I Use a Computer
(not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.Possible Solutions?
I Play the Violin instead
I Use a Computer (not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Problem: There is NO geometric constructionusing a straight-edge and compassthat will construct a length of 12
√.5.
This is a consequence of Galois Theory,we discuss this in senior-level algebra courses.It is equivalent to the problem of ’duplicating the cube’:Given a cube, construct a cube of exactly twice the volume.Possible Solutions?
I Play the Violin instead
I Use a Computer (not available in the Renaissance)
I Approximate somehow
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Strahle’s Construction (exposed by Barbour1957)
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Lay out a segment QR of length 12
I Construct an isosceles triangle OQR with sides of length24
I Fix the point P on OQ such that PQ has length 7
I Draw the line RP and the point M on that line with MP= RP(the guitar, with neck at R, bridge at M, and octave atP)
I Fret the guitar at the intersections of MR with the linesthrough O meeting QR at the 12 points dividing QRequally.
Could this work?
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Lay out a segment QR of length 12
I Construct an isosceles triangle OQR with sides of length24
I Fix the point P on OQ such that PQ has length 7
I Draw the line RP and the point M on that line with MP= RP(the guitar, with neck at R, bridge at M, and octave atP)
I Fret the guitar at the intersections of MR with the linesthrough O meeting QR at the 12 points dividing QRequally.
Could this work?
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Lay out a segment QR of length 12
I Construct an isosceles triangle OQR with sides of length24
I Fix the point P on OQ such that PQ has length 7
I Draw the line RP and the point M on that line with MP= RP(the guitar, with neck at R, bridge at M, and octave atP)
I Fret the guitar at the intersections of MR with the linesthrough O meeting QR at the 12 points dividing QRequally.
Could this work?
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Lay out a segment QR of length 12
I Construct an isosceles triangle OQR with sides of length24
I Fix the point P on OQ such that PQ has length 7
I Draw the line RP and the point M on that line with MP= RP(the guitar, with neck at R, bridge at M, and octave atP)
I Fret the guitar at the intersections of MR with the linesthrough O meeting QR at the 12 points dividing QRequally.
Could this work?
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Lay out a segment QR of length 12
I Construct an isosceles triangle OQR with sides of length24
I Fix the point P on OQ such that PQ has length 7
I Draw the line RP and the point M on that line with MP= RP(the guitar, with neck at R, bridge at M, and octave atP)
I Fret the guitar at the intersections of MR with the linesthrough O meeting QR at the 12 points dividing QRequally.
Could this work?
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Lay out a segment QR of length 12
I Construct an isosceles triangle OQR with sides of length24
I Fix the point P on OQ such that PQ has length 7
I Draw the line RP and the point M on that line with MP= RP(the guitar, with neck at R, bridge at M, and octave atP)
I Fret the guitar at the intersections of MR with the linesthrough O meeting QR at the 12 points dividing QRequally.
Could this work?
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Mathematics of Projections
Suppose you have two lines L1 and L2 in the plane and apoint O not on either line.
Then one has a correspondence between the points of L1and L2 given by ”projection” from O.The projection π : L1 → L2 is defined geometrically, but aformula for π can be obtained if one has coordinate systemson the two lines.Indeed, if x is a coordinate on L1 and y is a coordinate on L2(with different origins, and different scales, allowed) then themapping π will send a point on L1 with coordinate x to apoint on L2 with coordinate y = y(x); and this functionalways has the form
y(x) =a + bx
c + dx
for suitable constants a, b, c, and d .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Mathematics of Projections
Suppose you have two lines L1 and L2 in the plane and apoint O not on either line.Then one has a correspondence between the points of L1and L2 given by ”projection” from O.
The projection π : L1 → L2 is defined geometrically, but aformula for π can be obtained if one has coordinate systemson the two lines.Indeed, if x is a coordinate on L1 and y is a coordinate on L2(with different origins, and different scales, allowed) then themapping π will send a point on L1 with coordinate x to apoint on L2 with coordinate y = y(x); and this functionalways has the form
y(x) =a + bx
c + dx
for suitable constants a, b, c, and d .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Mathematics of Projections
Suppose you have two lines L1 and L2 in the plane and apoint O not on either line.Then one has a correspondence between the points of L1and L2 given by ”projection” from O.The projection π : L1 → L2 is defined geometrically, but aformula for π can be obtained if one has coordinate systemson the two lines.
Indeed, if x is a coordinate on L1 and y is a coordinate on L2(with different origins, and different scales, allowed) then themapping π will send a point on L1 with coordinate x to apoint on L2 with coordinate y = y(x); and this functionalways has the form
y(x) =a + bx
c + dx
for suitable constants a, b, c, and d .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Mathematics of Projections
Suppose you have two lines L1 and L2 in the plane and apoint O not on either line.Then one has a correspondence between the points of L1and L2 given by ”projection” from O.The projection π : L1 → L2 is defined geometrically, but aformula for π can be obtained if one has coordinate systemson the two lines.Indeed, if x is a coordinate on L1 and y is a coordinate on L2(with different origins, and different scales, allowed) then themapping π will send a point on L1 with coordinate x to apoint on L2 with coordinate y = y(x); and this functionalways has the form
y(x) =a + bx
c + dx
for suitable constants a, b, c, and d .
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
For Strahle’s construction, if you have a coordinate x on thesegment QR which is 0 at R and 1 at Q, and a coordinate yon the guitar which is 0 at M and 1 at R, then theprojection function is
y =17− 5x17 + 7x
.
This gives the lengths for the notes as:
Note C D E F G A B CStrahle 1 .8899 .7931 .7490 .6680 .5955 .5302 .5Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5
Percent 0 −0.111 −0.075 0.027 0.085 0.15 0.098 0
Pretty darn good!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
For Strahle’s construction, if you have a coordinate x on thesegment QR which is 0 at R and 1 at Q, and a coordinate yon the guitar which is 0 at M and 1 at R, then theprojection function is
y =17− 5x17 + 7x
.
This gives the lengths for the notes as:
Note C D E F G A B CStrahle 1 .8899 .7931 .7490 .6680 .5955 .5302 .5Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5
Percent 0 −0.111 −0.075 0.027 0.085 0.15 0.098 0
Pretty darn good!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
For Strahle’s construction, if you have a coordinate x on thesegment QR which is 0 at R and 1 at Q, and a coordinate yon the guitar which is 0 at M and 1 at R, then theprojection function is
y =17− 5x17 + 7x
.
This gives the lengths for the notes as:
Note C D E F G A B CStrahle 1 .8899 .7931 .7490 .6680 .5955 .5302 .5Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5
Percent 0 −0.111 −0.075 0.027 0.085 0.15 0.098 0
Pretty darn good!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Why is (17− 5x)/(17 + 7x) so good?I How did Strahle think of this?
����������
CCCCCCCCCC
@@
@@@
@@@
Q N S R
T
P
M
O
7
17
I PQN similar to OQS; hence |QN|/7 = |QS |/24 or|QN| = 724 ∗ |QS | =
748 ∗ |QR|.
I Hence |NR| = 4148 ∗ |QR|; and|SR|/|NR| = 1/241/48 = 24/41.
I PNR similar to TSR; hence |TR|/|SR| = |PR|/|NR|I |PR| = |MR|/2; Hence
|TR| = |PR| ∗ (|SR|/|NR|) = (12/41) ∗ |MR|.I Therefore |MT | = (29/41) ∗ |MR|.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I Why is (17− 5x)/(17 + 7x) so good?I How did Strahle think of this?
����������
CCCCCCCCCC
@@
@@@
@@@
Q N S R
T
P
M
O
7
17
I PQN similar to OQS; hence |QN|/7 = |QS |/24 or|QN| = 724 ∗ |QS | =
748 ∗ |QR|.
I Hence |NR| = 4148 ∗ |QR|; and|SR|/|NR| = 1/241/48 = 24/41.
I PNR similar to TSR; hence |TR|/|SR| = |PR|/|NR|I |PR| = |MR|/2; Hence
|TR| = |PR| ∗ (|SR|/|NR|) = (12/41) ∗ |MR|.I Therefore |MT | = (29/41) ∗ |MR|.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Most Accurate ProjectionSuppose you look for a projection function
y(x) =a + bx
c + dx
which is gives the most accurate lengths for the notes.
This means you’d want constants a, b, c , and d such that
a + bx
c + dx≈ (.5)x
and your frets could then be placed by substituting x = 0,1/12, 2/12, . . . , 11/12, 1 into the linear fractional formula.This appears to be four degrees of freedom, but is actuallyonly three. (Only the ratios of the a,b,c,d count.)You would have to have
y(0) = 1 and y(1) = 1/2
in order to fix the neck and the octave exactly.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Most Accurate ProjectionSuppose you look for a projection function
y(x) =a + bx
c + dx
which is gives the most accurate lengths for the notes.This means you’d want constants a, b, c , and d such that
a + bx
c + dx≈ (.5)x
and your frets could then be placed by substituting x = 0,1/12, 2/12, . . . , 11/12, 1 into the linear fractional formula.
This appears to be four degrees of freedom, but is actuallyonly three. (Only the ratios of the a,b,c,d count.)You would have to have
y(0) = 1 and y(1) = 1/2
in order to fix the neck and the octave exactly.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Most Accurate ProjectionSuppose you look for a projection function
y(x) =a + bx
c + dx
which is gives the most accurate lengths for the notes.This means you’d want constants a, b, c , and d such that
a + bx
c + dx≈ (.5)x
and your frets could then be placed by substituting x = 0,1/12, 2/12, . . . , 11/12, 1 into the linear fractional formula.This appears to be four degrees of freedom, but is actuallyonly three. (Only the ratios of the a,b,c,d count.)
You would have to have
y(0) = 1 and y(1) = 1/2
in order to fix the neck and the octave exactly.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
The Most Accurate ProjectionSuppose you look for a projection function
y(x) =a + bx
c + dx
which is gives the most accurate lengths for the notes.This means you’d want constants a, b, c , and d such that
a + bx
c + dx≈ (.5)x
and your frets could then be placed by substituting x = 0,1/12, 2/12, . . . , 11/12, 1 into the linear fractional formula.This appears to be four degrees of freedom, but is actuallyonly three. (Only the ratios of the a,b,c,d count.)You would have to have
y(0) = 1 and y(1) = 1/2
in order to fix the neck and the octave exactly.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
If you try to place the half-way note (’Tritone’) exactly, youwould then need
a + b/2
c + d/2= (.5)6/12 =
√.5 = 1/
√2.
Solving these three equations for a, b, c , and d (andremembering that only the ratios count) leads to the bestapproximate projection function:
(.5)x ≈ (2−√
2) + (2√
2− 3)x(2−
√2) + (3
√2− 4)x
.
Yucchh!This is NOT what Strahle came up with, and it is not likelythat a simple geometric construction like his would find thisexact projection.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
If you try to place the half-way note (’Tritone’) exactly, youwould then need
a + b/2
c + d/2= (.5)6/12 =
√.5 = 1/
√2.
Solving these three equations for a, b, c , and d (andremembering that only the ratios count) leads to the bestapproximate projection function:
(.5)x ≈ (2−√
2) + (2√
2− 3)x(2−
√2) + (3
√2− 4)x
.
Yucchh!This is NOT what Strahle came up with, and it is not likelythat a simple geometric construction like his would find thisexact projection.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
If you try to place the half-way note (’Tritone’) exactly, youwould then need
a + b/2
c + d/2= (.5)6/12 =
√.5 = 1/
√2.
Solving these three equations for a, b, c , and d (andremembering that only the ratios count) leads to the bestapproximate projection function:
(.5)x ≈ (2−√
2) + (2√
2− 3)x(2−
√2) + (3
√2− 4)x
.
Yucchh!
This is NOT what Strahle came up with, and it is not likelythat a simple geometric construction like his would find thisexact projection.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
If you try to place the half-way note (’Tritone’) exactly, youwould then need
a + b/2
c + d/2= (.5)6/12 =
√.5 = 1/
√2.
Solving these three equations for a, b, c , and d (andremembering that only the ratios count) leads to the bestapproximate projection function:
(.5)x ≈ (2−√
2) + (2√
2− 3)x(2−
√2) + (3
√2− 4)x
.
Yucchh!This is NOT what Strahle came up with, and it is not likelythat a simple geometric construction like his would find thisexact projection.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Continued Fraction Approximations of Numbers
Strahle’s formula
y(x) =17− 5x17 + 7x
satisfies
y(0) = 1, y(1) = 1/2, but y(1/2) =17− 5/217 + 7/2
=34− 534 + 7
=29
41.
29
416=√.5 = 1/
√2 since
41
296=√
2.
Indeed, there is no rational number p/q such that p/q =√
2;squaring both sides and multiplying by q2 would give
p2 = 2q2
and this can’t be true if p and q have no common factors.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Continued Fraction Approximations of Numbers
Strahle’s formula
y(x) =17− 5x17 + 7x
satisfies
y(0) = 1, y(1) = 1/2, but y(1/2) =17− 5/217 + 7/2
=34− 534 + 7
=29
41.
29
416=√.5 = 1/
√2 since
41
296=√
2.
Indeed, there is no rational number p/q such that p/q =√
2;squaring both sides and multiplying by q2 would give
p2 = 2q2
and this can’t be true if p and q have no common factors.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Continued Fraction Approximations of Numbers
Strahle’s formula
y(x) =17− 5x17 + 7x
satisfies
y(0) = 1, y(1) = 1/2, but y(1/2) =17− 5/217 + 7/2
=34− 534 + 7
=29
41.
29
416=√.5 = 1/
√2 since
41
296=√
2.
Indeed, there is no rational number p/q such that p/q =√
2;squaring both sides and multiplying by q2 would give
p2 = 2q2
and this can’t be true if p and q have no common factors.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
But there are numbers such that
p2 − 2q2 = ±1, ”Pell’s Equation”
the closest one could get.
Note that
412 = 1681, 292 = 841, 2∗292 = 1682, 412−2∗292 = −1
so 41/29 is a great approximation to√
2.Indeed,
41
29= 1.413793103 and
√2 = 1.414213562
41
29= 1 +
1
2 + 12+ 1
2+ 12
which is the truncation of the full continued fractionexpansion of
√2, and all truncations give all solutions to
Pell’s Equation, and the best rational approximations to√
2.There is no evidence at all that Strahle knew any of this!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
But there are numbers such that
p2 − 2q2 = ±1, ”Pell’s Equation”
the closest one could get.Note that
412 = 1681, 292 = 841, 2∗292 = 1682, 412−2∗292 = −1
so 41/29 is a great approximation to√
2.
Indeed,
41
29= 1.413793103 and
√2 = 1.414213562
41
29= 1 +
1
2 + 12+ 1
2+ 12
which is the truncation of the full continued fractionexpansion of
√2, and all truncations give all solutions to
Pell’s Equation, and the best rational approximations to√
2.There is no evidence at all that Strahle knew any of this!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
But there are numbers such that
p2 − 2q2 = ±1, ”Pell’s Equation”
the closest one could get.Note that
412 = 1681, 292 = 841, 2∗292 = 1682, 412−2∗292 = −1
so 41/29 is a great approximation to√
2.Indeed,
41
29= 1.413793103 and
√2 = 1.414213562
41
29= 1 +
1
2 + 12+ 1
2+ 12
which is the truncation of the full continued fractionexpansion of
√2, and all truncations give all solutions to
Pell’s Equation, and the best rational approximations to√
2.There is no evidence at all that Strahle knew any of this!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
But there are numbers such that
p2 − 2q2 = ±1, ”Pell’s Equation”
the closest one could get.Note that
412 = 1681, 292 = 841, 2∗292 = 1682, 412−2∗292 = −1
so 41/29 is a great approximation to√
2.Indeed,
41
29= 1.413793103 and
√2 = 1.414213562
41
29= 1 +
1
2 + 12+ 1
2+ 12
which is the truncation of the full continued fractionexpansion of
√2, and all truncations give all solutions to
Pell’s Equation, and the best rational approximations to√
2.There is no evidence at all that Strahle knew any of this!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
But there are numbers such that
p2 − 2q2 = ±1, ”Pell’s Equation”
the closest one could get.Note that
412 = 1681, 292 = 841, 2∗292 = 1682, 412−2∗292 = −1
so 41/29 is a great approximation to√
2.Indeed,
41
29= 1.413793103 and
√2 = 1.414213562
41
29= 1 +
1
2 + 12+ 1
2+ 12
which is the truncation of the full continued fractionexpansion of
√2, and all truncations give all solutions to
Pell’s Equation, and the best rational approximations to√
2.
There is no evidence at all that Strahle knew any of this!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
But there are numbers such that
p2 − 2q2 = ±1, ”Pell’s Equation”
the closest one could get.Note that
412 = 1681, 292 = 841, 2∗292 = 1682, 412−2∗292 = −1
so 41/29 is a great approximation to√
2.Indeed,
41
29= 1.413793103 and
√2 = 1.414213562
41
29= 1 +
1
2 + 12+ 1
2+ 12
which is the truncation of the full continued fractionexpansion of
√2, and all truncations give all solutions to
Pell’s Equation, and the best rational approximations to√
2.There is no evidence at all that Strahle knew any of this!
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Galilei’s Approximation
Vincenzo Galilei:the father of the famous astronomer Galileo Galilei
He suggested using
Half-note length ratio =17
18= .944444444 · · ·
(Pyth. =243
256= .94921875 · · · and Equal = .943874313 · · · )
(18/17 is the first continued fraction approximation to 12√
2.)If you compute, you find that
(17
18)12 = 0.503636 · · ·
so the Octave is off by .003636, a bit too short.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Galilei’s Approximation
Vincenzo Galilei:the father of the famous astronomer Galileo GalileiHe suggested using
Half-note length ratio =17
18= .944444444 · · ·
(Pyth. =243
256= .94921875 · · · and Equal = .943874313 · · · )
(18/17 is the first continued fraction approximation to 12√
2.)If you compute, you find that
(17
18)12 = 0.503636 · · ·
so the Octave is off by .003636, a bit too short.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Galilei’s Approximation
Vincenzo Galilei:the father of the famous astronomer Galileo GalileiHe suggested using
Half-note length ratio =17
18= .944444444 · · ·
(Pyth. =243
256= .94921875 · · · and Equal = .943874313 · · · )
(18/17 is the first continued fraction approximation to 12√
2.)If you compute, you find that
(17
18)12 = 0.503636 · · ·
so the Octave is off by .003636, a bit too short.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Galilei’s Approximation
Vincenzo Galilei:the father of the famous astronomer Galileo GalileiHe suggested using
Half-note length ratio =17
18= .944444444 · · ·
(Pyth. =243
256= .94921875 · · · and Equal = .943874313 · · · )
(18/17 is the first continued fraction approximation to 12√
2.)
If you compute, you find that
(17
18)12 = 0.503636 · · ·
so the Octave is off by .003636, a bit too short.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Galilei’s Approximation
Vincenzo Galilei:the father of the famous astronomer Galileo GalileiHe suggested using
Half-note length ratio =17
18= .944444444 · · ·
(Pyth. =243
256= .94921875 · · · and Equal = .943874313 · · · )
(18/17 is the first continued fraction approximation to 12√
2.)If you compute, you find that
(17
18)12 = 0.503636 · · ·
so the Octave is off by .003636, a bit too short.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Vincenzo’s solution was to just shorten the total string then,by exactly the amount to make this point (0.503636)halfway.
This point is 1− .503636 = .496363734 down the string, soyou double it to get 0.992727468, cutting of 0.007272531 ofthe string.Mathematically, this makes the Nth note in the scale havelength
(17/18)N − .0072725310.992727468
giving the lengths indicated below:
Note C D E F G A B CVincenzo 1 .8912 .7941 .7496 .6678 .5949 .5298 .5
Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5Percent 0 0.032 0.053 0.059 0.062 0.049 0.021 0
This is a fabulous approximation!His discovery that the pitch created by a string variednonlinearly with the tension was one of the first non-linearphysical laws discovered.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Vincenzo’s solution was to just shorten the total string then,by exactly the amount to make this point (0.503636)halfway.This point is 1− .503636 = .496363734 down the string, soyou double it to get 0.992727468, cutting of 0.007272531 ofthe string.
Mathematically, this makes the Nth note in the scale havelength
(17/18)N − .0072725310.992727468
giving the lengths indicated below:
Note C D E F G A B CVincenzo 1 .8912 .7941 .7496 .6678 .5949 .5298 .5
Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5Percent 0 0.032 0.053 0.059 0.062 0.049 0.021 0
This is a fabulous approximation!His discovery that the pitch created by a string variednonlinearly with the tension was one of the first non-linearphysical laws discovered.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Vincenzo’s solution was to just shorten the total string then,by exactly the amount to make this point (0.503636)halfway.This point is 1− .503636 = .496363734 down the string, soyou double it to get 0.992727468, cutting of 0.007272531 ofthe string.Mathematically, this makes the Nth note in the scale havelength
(17/18)N − .0072725310.992727468
giving the lengths indicated below:
Note C D E F G A B CVincenzo 1 .8912 .7941 .7496 .6678 .5949 .5298 .5
Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5Percent 0 0.032 0.053 0.059 0.062 0.049 0.021 0
This is a fabulous approximation!
His discovery that the pitch created by a string variednonlinearly with the tension was one of the first non-linearphysical laws discovered.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
Vincenzo’s solution was to just shorten the total string then,by exactly the amount to make this point (0.503636)halfway.This point is 1− .503636 = .496363734 down the string, soyou double it to get 0.992727468, cutting of 0.007272531 ofthe string.Mathematically, this makes the Nth note in the scale havelength
(17/18)N − .0072725310.992727468
giving the lengths indicated below:
Note C D E F G A B CVincenzo 1 .8912 .7941 .7496 .6678 .5949 .5298 .5
Equal 1 .8909 .7937 .7492 .6674 .5946 .5297 .5Percent 0 0.032 0.053 0.059 0.062 0.049 0.021 0
This is a fabulous approximation!His discovery that the pitch created by a string variednonlinearly with the tension was one of the first non-linearphysical laws discovered.
-
Musical, Physical,and Mathematical
IntervalsThe 2010 Leonard
Sulski LectureCollege of the Holy
Cross
Rick Miranda
The Physics ofSound
Length (orFrequency) RatiosBetween Notes
Fretting A Guitar
GeometricalApproximations
ArithmeticApproximations
Vincenzo Galilei
I J.M. Barbour: A geometrical approximation to theroots of numbers. American Mathematical Monthly,Vol. 64, No. 1 (1957), 1–9.
I V. Galilei: Dialogo della musica antica e moderna,Florence (1581), p. 49
I D.P. Strahle: Nytt pafund, til at finna temperaturen istamningen for thonerne pa claveret ock dylikainstrumeter Proceedings of the Swedish Academy(1743), Vol. IV, 281–291
I Ian Stewart
The Physics of SoundLength (or Frequency) Ratios Between NotesFretting A GuitarGeometrical ApproximationsArithmetic ApproximationsVincenzo Galilei